On 1 Maj, 23:31, Dan <dan.ms.ch...@gmail.com> wrote: > On May 1, 11:18 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > > > > > On 1 Mai, 18:27, Dan <dan.ms.ch...@gmail.com> wrote: > > > > They have similar formulas,but behave in different ways . > > > You would be correct in affirming that b_inf = 1/9 is not part of the > > > list . > > > Of course. It is not part of the list, because it cannot be written as > > decimal number. Proof: All decimal numbers that can be written in this > > form *are already in the list*. And there is no reason why 1/9 should > > be missing, if it could be a decimal fraction. > > > > If it were part of the list , then a_inf would be a well-behaved > > > natural number , but it isn't . > > > It is neither well behaved nor a natural number. The sequence 0.111... > > does not exist other than by a finite name or formula. > > > > Limits only work properly with real numbers, not natural numbers . > > > Yes, that is true. But (and please read this very attentively!): > > Cantor's argument requires the existence of the complete sequence > > 0.111.... in digits: > > > You can see this easily here: > > > The list > > > 0.0 > > 0.1 > > 0.11 > > 0.111 > > ... > > > when replacing 0 by 1 has an anti-diagonal, the FIS of which are > > always in the next line. So the anti-diagonal is not different from > > all lines, unless it has an infinite sequence of 1's. But, as we just > > saw, this is impossible. > > > Regards, WM > > I see no significant difference between referring to a mathematical > object by a formula and referring to it by 'writing it down' . > Writing it down , when possible , is just another formula for it. > "1296" , and "36 * 36" , are both references to the same object , > and neither of them is "more true of a name" for the object than the > other . > > Just like "1296" is "36 * 36" , and we can substitute one for another > in mathematical expressions , and maintain their truth , > so is "1/9" the same as "0. the infinite sequence of 1's" . That we > cannot truly write down the name of the second expression is of no > relevance. The most important point is that 'names for the same > object , whether finite or infinite, are fully interchangeable' > > Since > "0. the infinite sequence of 1's" is the same as "1/9" then > > "The third digit of 0. the infinite sequence of 1's" is the same as > "The third digit of 1/9" is the same as "1" . > > Same object . Different names .Since we treat all possible names of > the object the same , whether it be "a formula for the object" , or an > "enumeration of its digits" , we never run into problems. And in every > name for an object we can recover every other name, if we so wish . > > "1+3" = " 2+2" = "3+1" = "8-4" = "2*2" = "4" = "00 ..... 04" > > "A rose by any other name would smell as sweet"
No no no a formula is not a number... A function is not a number... An expression can be evaluated into a number. But 8 is a number 2*4 is an expression 4+4 is an expression 2^3 is an expression they are all different expressions/calculations leading to same value.